Solve by Factoring 2/(x-2)-4/(x^2-2x)=5

$\frac{2}{x - 2} - \frac{4}{x^{2} - 2 x} = 5$

Step 1

Subtract $5$ from both sides of the equation.

$\frac{2}{x - 2} - \frac{4}{x^{2} - 2 x} - 5 = 0$

Step 2

Simplify $\frac{2}{x - 2} - \frac{4}{x^{2} - 2 x} - 5$ .

Factor $x$ out of $x^{2} - 2 x$ .

Factor $x$ out of $x^{2}$ .

$\frac{2}{x - 2} - \frac{4}{x \cdot x - 2 x} - 5 = 0$

Factor $x$ out of $- 2 x$ .

$\frac{2}{x - 2} - \frac{4}{x \cdot x + x \cdot - 2} - 5 = 0$

Factor $x$ out of $x \cdot x + x \cdot - 2$ .

$\frac{2}{x - 2} - \frac{4}{x (x - 2)} - 5 = 0$

To write $\frac{2}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{2}{x - 2} \cdot \frac{x}{x} - \frac{4}{x (x - 2)} - 5 = 0$

Write each expression with a common denominator of $(x - 2) x$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{2}{x - 2}$ by $\frac{x}{x}$ .

$\frac{2 x}{(x - 2) x} - \frac{4}{x (x - 2)} - 5 = 0$

Reorder the factors of $(x - 2) x$ .

$\frac{2 x}{x (x - 2)} - \frac{4}{x (x - 2)} - 5 = 0$

Combine the numerators over the common denominator.

$\frac{2 x - 4}{x (x - 2)} - 5 = 0$

Simplify each term.

Factor $2$ out of $2 x - 4$ .

Factor $2$ out of $2 x$ .

$\frac{2 (x) - 4}{x (x - 2)} - 5 = 0$

Factor $2$ out of $- 4$ .

$\frac{2 x + 2 \cdot - 2}{x (x - 2)} - 5 = 0$

Factor $2$ out of $2 x + 2 \cdot - 2$ .

$\frac{2 (x - 2)}{x (x - 2)} - 5 = 0$

Cancel the common factor of $x - 2$ .

Cancel the common factor.

$\frac{2 (x - 2)}{x (x - 2)} - 5 = 0$

Rewrite the expression.

$\frac{2}{x} - 5 = 0$

Step 3

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{2}{x} - 5 \cdot \frac{x}{x} = 0$

Step 4

Combine $- 5$ and $\frac{x}{x}$ .

$\frac{2}{x} + \frac{- 5 x}{x} = 0$

Step 5

Combine the numerators over the common denominator.

$\frac{2 - 5 x}{x} = 0$

Step 6

Set the numerator equal to zero.

$2 - 5 x = 0$

Step 7

Solve the equation for $x$ .

Subtract $2$ from both sides of the equation.

$- 5 x = - 2$

Divide each term in $- 5 x = - 2$ by $- 5$ and simplify.

Divide each term in $- 5 x = - 2$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{- 2}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{- 2}{- 5}$

Divide $x$ by $1$ .

$x = \frac{- 2}{- 5}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x = \frac{2}{5}$

Solve by Factoring 2/(x-2)-4/(x^2-2x)=5

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